Questions — OCR MEI (4456 questions)

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OCR MEI Paper 2 2018 June Q9
5 marks Easy -1.8
9 At the end of each school term at North End College all the science classes in year 10 are given a test. The marks out of 100 achieved by members of set 1 are shown in Fig. 9. \begin{table}[h]
35
409
5236
601356
701256899
83466889
955567
\captionsetup{labelformat=empty} \caption{Fig. 9}
\end{table} Key \(5 \quad\) 2 represents a mark of 52
  1. Describe the shape of the distribution.
  2. The teacher for set 1 claimed that a typical student in his class achieved a mark of 95. How did he justify this statement?
  3. Another teacher said that the average mark in set 1 is 76 . How did she justify this statement? Benson's mark in the test is 35 . If the mark achieved by any student is an outlier in the lower tail of the distribution, the student is moved down to set 2 .
  4. Determine whether Benson is moved down to set 2 .
OCR MEI Paper 2 2018 June Q10
8 marks Standard +0.3
10 The screenshot in Fig. 10 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-06_515_1009_338_529} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} The area of each of the unshaded regions under the curve is 0.025 . The lower boundary of the shaded region is at 16.452 and the upper boundary of the shaded region is at 21.548 .
  1. Calculate the value of \(\mu\).
  2. Calculate the value of \(\sigma ^ { 2 }\).
  3. \(Y\) is the random variable given by \(Y = 4 X + 5\).
    (A) Write down the distribution of \(Y\).
    (B) Find \(\mathrm { P } ( \mathrm { Y } > 90 )\).
OCR MEI Paper 2 2018 June Q11
6 marks Moderate -0.8
11 The discrete random variable \(X\) takes the values \(0,1,2,3,4\) and 5 with probabilities given by the formula $$\mathrm { P } ( X = x ) = k ( x + 1 ) ( 6 - x ) .$$
  1. Find the value of \(k\). In one half-term Ben attends school on 40 days. The probability distribution above is used to model \(X\), the number of lessons per day in which Ben receives a gold star for excellent work.
  2. Find the probability that Ben receives no gold stars on each of the first 3 days of the half-term and two gold stars on each of the next 2 days.
  3. Find the expected number of days in the half-term on which Ben receives no gold stars.
OCR MEI Paper 2 2018 June Q13
10 marks Challenging +1.2
13 Each weekday Keira drives to work with her son Kaito. She always sets off at 8.00 a.m. She models her journey time, \(x\) minutes, by the distribution \(X \sim \mathrm {~N} ( 15,4 )\). Over a long period of time she notes that her journey takes less than 14 minutes on \(7 \%\) of the journeys, and takes more than 18 minutes on \(31 \%\) of the journeys.
  1. Investigate whether Keira's model is a good fit for the data. Kaito believes that Keira's value for the variance is correct, but realises that the mean is not correct.
  2. Find, correct to two significant figures, the value of the mean that Keira should use in a refined model which does fit the data. Keira buys a new car. After driving to work in it each day for several weeks, she randomly selects the journey times for \(n\) of these days. Her mean journey time for these \(n\) days is 16 minutes. Using the refined model she conducts a hypothesis test to see if her mean journey time has changed, and finds that the result is significant at the \(5 \%\) level.
  3. Determine the smallest possible value of \(n\).
OCR MEI Paper 2 2018 June Q14
9 marks Moderate -0.8
14 The pre-release material includes data on unemployment rates in different countries. A sample from this material has been taken. All the countries in the sample are in Europe. The data have been grouped and are shown in Fig 14.1. \begin{table}[h]
Unemployment rate\(0 -\)\(5 -\)\(10 -\)\(15 -\)\(20 -\)\(35 - 50\)
Frequency15215522
\captionsetup{labelformat=empty} \caption{Fig. 14.1}
\end{table} A cumulative frequency curve has been generated for the sample data using a spreadsheet. This is shown in Fig. 14.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-08_639_1081_808_466} \captionsetup{labelformat=empty} \caption{Fig. 14.2}
\end{figure} Hodge used Fig. 14.2 to estimate the median unemployment rate in Europe. He obtained the answer 5.0. The correct value for this sample is 6.9.
  1. (A) There is a systematic error in the diagram.
    The scatter diagram shown in Fig. 14.3 shows the unemployment rate and life expectancy at birth for the 47 countries in the sample for which this information is available. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Scatter diagram to show life expectancy at birth against unemployment rate} \includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-09_627_1281_456_367}
    \end{figure} Fig. 14.3 The product moment correlation coefficient for the 47 items in the sample is - 0.2607 .
    The \(p\)-value associated with \(r = - 0.2607\) and \(n = 47\) is 0.0383 .
  2. Does this information suggest that there is an association between unemployment rate and life expectancy at birth in countries in Europe? Hodge uses the spreadsheet tools to obtain the equation of a line of best fit for this data.
  3. The unemployment rate in Kosovo is 35.3 , but there is no data available on life expectancy. Is it reasonable to use Hodge's line of best fit to estimate life expectancy at birth in Kosovo?
OCR MEI Paper 2 2018 June Q15
9 marks Standard +0.8
15 You must show detailed reasoning in this question. The equation of a curve is $$y ^ { 3 } - x y + 4 \sqrt { x } = 4 .$$ Find the gradient of the curve at each of the points where \(y = 1\).
OCR MEI Paper 2 2018 June Q16
11 marks Standard +0.3
16 In the first year of a course, an A-level student, Aaishah, has a mathematics test each week. The night before each test she revises for \(t\) hours. Over the course of the year she realises that her percentage mark for a test, \(p\), may be modelled by the following formula, where \(A , B\) and \(C\) are constants. $$p = A - B ( t - C ) ^ { 2 }$$
  • Aaishah finds that, however much she revises, her maximum mark is achieved when she does 2 hours revision. This maximum mark is 62 .
  • Aaishah had a mark of 22 when she didn't spend any time revising.
    1. Find the values of \(A , B\) and \(C\).
    2. According to the model, if Aaishah revises for 45 minutes on the night before the test, what mark will she achieve?
    3. What is the maximum amount of time that Aaishah could have spent revising for the model to work?
In an attempt to improve her marks Aaishah now works through problems for a total of \(t\) hours over the three nights before the test. After taking a number of tests, she proposes the following new formula for \(p\). $$p = 22 + 68 \left( 1 - \mathrm { e } ^ { - 0.8 t } \right)$$ For the next three tests she recorded the data in Fig. 16. \begin{table}[h]
\(t\)135
\(p\)598489
\captionsetup{labelformat=empty} \caption{Fig. 16}
\end{table}
  • Verify that the data is consistent with the new formula.
  • Aaishah's tutor advises her to spend a minimum of twelve hours working through problems in future. Determine whether or not this is good advice.
    •
    OCR MEI Paper 2 2018 June Q17
    12 marks Standard +0.3
    17
    1. Express \(\frac { \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } }\) in partial fractions.
    2. Express \(y\) in terms of \(x\) given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y \left( x ^ { 2 } - 8 x + 9 \right) } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \text { and } y = 16 \text { when } x = 3 .$$ \section*{END OF QUESTION PAPER}
    OCR MEI Paper 2 2019 June Q1
    4 marks Easy -1.8
    1 Fig. 1 shows the probability distribution of the discrete random variable \(X\). \begin{table}[h]
    \(x\)12345
    \(\mathrm { P } ( X = x )\)0.20.1\(k\)\(2 k\)\(4 k\)
    \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{table}
    1. Find the value of \(k\).
    2. Find \(\mathrm { P } ( X \neq 4 )\).
    OCR MEI Paper 2 2019 June Q2
    4 marks Moderate -0.8
    2 Given that \(y = \left( x ^ { 2 } + 5 \right) ^ { 12 }\),
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence find \(\int 48 x \left( x ^ { 2 } + 5 \right) ^ { 11 } \mathrm {~d} x\).
    OCR MEI Paper 2 2019 June Q3
    3 marks Moderate -0.8
    3 Fig. 3 shows the time Lorraine spent in hours, \(t\), answering e-mails during the working day. The data were collected over a number of months. \begin{table}[h]
    Time in hours,
    \(t\)
    		\(0 \leqslant t < 1\)\(1 \leqslant t < 2\)\(2 \leqslant t < 3\)\(3 \leqslant t < 4\)\(4 \leqslant t < 6\)\(6 \leqslant t < 8\)
    Number of
    days
    		283642312412
    \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{table}
    1. Calculate an estimate of the mean time per day that Lorraine spent answering e-mails over this period.
    2. Explain why your answer to part (a) is an estimate. When Lorraine accepted her job, she was told that the mean time per day spent answering e-mails would not be more than 3 hours.
    3. Determine whether, according to the data in Fig. 3, it is possible that the mean time per day Lorraine spends answering e-mails is in fact more than 3 hours.
    OCR MEI Paper 2 2019 June Q4
    4 marks Moderate -0.3
    4 Fig. 4 shows the graph of \(y = \sqrt { 1 + x ^ { 3 } }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-05_544_639_338_248} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
    1. Use the trapezium rule with \(h = 0.5\) to find an estimate of \(\int _ { - 1 } ^ { 0 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x\), giving your answer correct to 6 decimal places.
    2. State whether your answer to part (a) is an under-estimate or an over-estimate, justifying your answer.
    OCR MEI Paper 2 2019 June Q5
    3 marks Easy -1.8
    5 Fig. 5 shows the number of times that students at a sixth form college visited a recreational mathematics website during the first week of the summer term. \begin{table}[h]
    Number of visits to website012345
    Number of students2438171242
    \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{table}
    1. State the value of the mid-range of the data.
    2. Describe the shape of the distribution.
    3. State the value of the mode.
    OCR MEI Paper 2 2019 June Q6
    4 marks Standard +0.3
    6 Find \(\int \frac { 32 } { x ^ { 5 } } \ln x \mathrm {~d} x\). Answer all the questions
    Section B (78 marks)
    OCR MEI Paper 2 2019 June Q7
    5 marks Standard +0.3
    7 The area of a sector of a circle is \(36.288 \mathrm {~cm} ^ { 2 }\). The angle of the sector is \(\theta\) radians and the radius of the circle is \(r \mathrm {~cm}\).
    1. Find an expression for \(\theta\) in terms of \(r\). The perimeter of the sector is 24.48 cm .
    2. Show that \(\theta = \frac { 24.48 } { r } - 2\).
    3. Find the possible values of \(r\).
    OCR MEI Paper 2 2019 June Q8
    9 marks Standard +0.3
    8 A team called "The Educated Guess" enter a weekly quiz. If they win the quiz in a particular week, the probability that they will win the following week is 0.4 , but if they do not win, the probability that they will win the following week is 0.2 . In week 4 The Educated Guess won the quiz.
    1. Calculate the probability that The Educated Guess will win the quiz in week 6. Every week the same 20 quiz teams, each with 6 members, take part in a quiz. Every member of every team buys a raffle ticket. Five winning tickets are drawn randomly, without replacement. Alf, who is a member of one of the teams, takes part every week.
    2. Calculate the probability that, in a randomly chosen week, Alf wins a raffle prize.
    3. Find the smallest number of weeks after which it will be \(95 \%\) certain that Alf has won at least one raffle prize.
    OCR MEI Paper 2 2019 June Q9
    9 marks Moderate -0.3
    9 You are given that \(\mathrm { f } ( x ) = 2 x + 3 \quad\) for \(x < 0 \quad\) and \(\mathrm { g } ( x ) = x ^ { 2 } - 2 x + 1\) for \(x > 1\).
    1. Find \(\mathrm { gf } ( x )\), stating the domain.
    2. State the range of \(\mathrm { gf } ( x )\).
    3. Find (gf) \({ } ^ { - 1 } ( x )\).
    OCR MEI Paper 2 2019 June Q10
    16 marks Moderate -0.3
    10 Club 65-80 Holidays fly jets between Liverpool and Magaluf. Over a long period of time records show that half of the flights from Liverpool to Magaluf take less than 153 minutes and \(5 \%\) of the flights take more than 183 minutes. An operations manager believes that flight times from Liverpool to Magaluf may be modelled by the Normal distribution.
    1. Use the information above to write down the mean time the operations manager will use in his Normal model for flight times from Liverpool to Magaluf.
    2. Use the information above to find the standard deviation the operations manager will use in his Normal model for flight times from Liverpool to Magaluf, giving your answer correct to 1 decimal place.
    3. Data is available for 452 flights. A flight time of under 2 hours was recorded in 16 of these flights. Use your answers to parts (a) and (b) to determine whether the model is consistent with this data. The operations manager suspects that the mean time for the journey from Magaluf to Liverpool is less than from Liverpool to Magaluf. He collects a random sample of 24 flight times from Magaluf to Liverpool. He finds that the mean flight time is 143.6 minutes.
    4. Use the Normal model used in part (c) to conduct a hypothesis test to determine whether there is evidence at the \(1 \%\) level to suggest that the mean flight time from Magaluf to Liverpool is less than the mean flight time from Liverpool to Magaluf.
      [0pt]
    5. Identify two ways in which the Normal model for flight times from Liverpool to Magaluf might be adapted to provide a better model for the flight times from Magaluf to Liverpool. [2]
    OCR MEI Paper 2 2019 June Q11
    8 marks Standard +0.3
    11 Fig. 11 shows the graph of \(y = x ^ { 2 } - 4 x + x \ln x\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-08_697_463_338_246} \captionsetup{labelformat=empty} \caption{Fig. 11}
    \end{figure}
    1. Show that the \(x\)-coordinate of the stationary point on the curve may be found from the equation \(2 x - 3 + \ln x = 0\).
    2. Use an iterative method to find the \(x\)-coordinate of the stationary point on the curve \(y = x ^ { 2 } - 4 x + x \ln x\), giving your answer correct to 4 decimal places.
    OCR MEI Paper 2 2019 June Q12
    10 marks Standard +0.8
    12 The jaguar is a species of big cat native to South America. Records show that 6\% of jaguars are born with black coats. Jaguars with black coats are known as black panthers. Due to deforestation a population of jaguars has become isolated in part of the Amazon basin. Researchers believe that the percentage of black panthers may not be \(6 \%\) in this population.
    1. Find the minimum sample size needed to conduct a two-tailed test to determine whether there is any evidence at the \(5 \%\) level to suggest that the percentage of black panthers is not \(6 \%\). A research team identifies 70 possible sites for monitoring the jaguars remotely. 30 of these sites are randomly selected and cameras are installed. 83 different jaguars are filmed during the evidence gathering period. The team finds that 10 of the jaguars are black panthers.
    2. Conduct a hypothesis test to determine whether the information gathered by the research team provides any evidence at the \(5 \%\) level to suggest that the percentage of black panthers in this population is not \(6 \%\).
    OCR MEI Paper 2 2019 June Q13
    7 marks Standard +0.3
    13 The population of Melchester is 185207. During a nationwide flu epidemic the number of new cases in Melchester are recorded each day. The results from the first three days are shown in Fig. 13. \begin{table}[h]
    Day123
    Number of new cases82472
    \captionsetup{labelformat=empty} \caption{Fig. 13}
    \end{table} A doctor notices that the numbers of new cases on successive days are in geometric progression.
    1. Find the common ratio for this geometric progression. The doctor uses this geometric progression to model the number of new cases of flu in Melchester.
    2. According to the model, how many new cases will there be on day 5?
    3. Find a formula for the total number of cases from day 1 to day \(n\) inclusive according to this model, simplifying your answer.
    4. Determine the maximum number of days for which the model could be viable in Melchester.
    5. State, with a reason, whether it is likely that the model will be viable for the number of days found in part (d).
    OCR MEI Paper 2 2019 June Q14
    9 marks Moderate -0.8
    14 The pre-release material includes data concerning crude death rates in different countries of the world. Fig. 14.1 shows some information concerning crude death rates in countries in Europe and in Africa. \begin{table}[h]
    EuropeAfrica
    \(n\)4856
    minimum6.283.58
    lower quartile8.507.31
    median9.538.71
    upper quartile11.4111.93
    maximum14.4614.89
    \captionsetup{labelformat=empty} \caption{Fig. 14.1}
    \end{table}
    1. Use your knowledge of the large data set to suggest a reason why the statistics in Fig. 14.1 refer to only 48 of the 51 European countries.
    2. Use the information in Fig. 14.1 to show that there are no outliers in either data set. The crude death rate in Libya is recorded as 3.58 and the population of Libya is recorded as 6411776.
    3. Calculate an estimate of the number of deaths in Libya in a year. The median age in Germany is 46.5 and the crude death rate is 11.42. The median age in Cyprus is 36.1 and the crude death rate is 6.62 .
    4. Explain why a country like Germany, with a higher median age than Cyprus, might also be expected to have a higher crude death rate than Cyprus. Fig. 14.2 shows a scatter diagram of median age against crude death rate for countries in Africa and Fig. 14.3 shows a scatter diagram of median age against crude death rate for countries in Europe. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-10_678_1221_1975_248} \captionsetup{labelformat=empty} \caption{Fig. 14.2}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_588_1248_223_228} \captionsetup{labelformat=empty} \caption{Fig. 14.3}
      \end{figure} The rank correlation coefficient for the data shown in Fig. 14.2 is - 0.281206 .
      The rank correlation coefficient for the data shown in Fig. 14.3 is 0.335215 .
    5. Compare and contrast what may be inferred about the relationship between median age and crude death rate in countries in Africa and in countries in Europe.
    OCR MEI Paper 2 2023 June Q1
    3 marks Easy -1.2
    1 Determine the sum of the infinite geometric series \(9 - 3 + 1 - \frac { 1 } { 3 } + \frac { 1 } { 9 } + \ldots\)
    OCR MEI Paper 2 2023 June Q2
    3 marks Easy -1.2
    2 The equation of a circle is \(x ^ { 2 } - 12 x + y ^ { 2 } + 8 y + 3 = 0\).
    1. Find the radius of the circle.
    2. State the coordinates of the centre of the circle.
    OCR MEI Paper 2 2023 June Q3
    3 marks Moderate -0.8
    3 In this question you must show detailed reasoning.
    Find the smallest possible positive integers \(m\) and \(n\) such that \(\left( \frac { 64 } { 49 } \right) ^ { - \frac { 3 } { 2 } } = \frac { m } { n }\).